RADBOUD UNIVERSITY NIJMEGEN

FACULTY OF SCIENCE

GROUPOID SYMMETRY AND CONSTRAINTS BRACKETOF GENERAL RELATIVITY REVISITED

M.SC. THESIS IN MATHEMATICAL PHYSICS

Author:Jan GŁOWACKI

Supervisor:Prof. Klaas LANDSMAN

Second reader:Dr. Annegret BURTSCHER

August 2019

Contents

I Mathematical framework 5

1 Diffeological spaces 6

1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Sheaf-theoretic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Categorical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Sheaves on sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.3 Category of diffeological spaces as a sheaf category . . . . . . . . . . . . . 18

1.2.4 Sheaf-theoretic origin of categorical properties . . . . . . . . . . . . . . . 24

1.3 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3.1 Internal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3.2 External . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.3.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Groupoids and algebroids 41

2.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Lie groupoids and Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Notes on diffeological Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . 47

II General Relativity 52

1 Dynamical approach to the Einstein equation 52

1.1 Initial value formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.2 Constraints bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 Diffeological algebroid of hypersurface deformations 56

2.1 Groupoid symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2 Diffeological groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 Group of bisections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4 Algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5 Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

1

Acknowledgements

I would like to express here my gratitude, firstly to my supervisor Prof. Klaas Landsman, forhis inspiration, encouragement and support, in all kinds of ways. I wish every ambitious studenta supervisor of this kind. Secondly, I wish to thank my family, without whose support the desirefor developing my passion for mathematical physics could not have been fulfilled: I am gratefulto my parents, Agnieszka and Paweł, and grandparents, Augustyna and Andrzej, for their continuoussupport. I appreciate my privileged position that allows me to put my time into investigating themysteries of Physics.

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IntroductionThe constraints that arise from the ADM formulation of the Einstein equation are agreed to be bothcrucially important and not fully understood. This thesis is a result of our struggle for understandingthe origin of their Poisson bracket relations that was proposed to emerge from a symmetry struc-ture inherent to the initial value formulation by Weinstein, Blohmann and Fernandes in the paper“Groupoid symmetry and constraints of General Relativity” [2]. We find this idea, as well as thesymmetry structure that they propose, very appealing. However, the analysis as presented there mightnot be fully satisfactory for the following reasons:

1. There exist no general framework for diffeological algebroids, which inevitably led to a bitsketchy character of some of the points of the analysis.

2. The diffeological structure that is put on the groupoid feels unnatural and does not makeit crystal clear that the bracket structure thus derived is really emergent from the groupoidstructure.

3. The resulting global structure of the algebroid, just like the general strategy of derivingit, seems to us to be more complex than necessary.

We aim to make some improvements at all the points mentioned above. The approach that we describein details is sketched in the Appendix of [2], all the ideas that we present are basically already there.We make some changes on the level of the diffeology and develop a detailed analysis through thegroup of bisections. We aim to make the derivation conceptually and technically clearer and moreconvincing which, in our view, was accomplished this way.

The first part of the thesis aims to improve the situation described in the first point above. We introduceall the mathematical frameworks that we will use, beginning with a deep, yet elementary analysisof the theory of diffeological spaces from the sheaf-theoretic point of view. This perspective is notnecessary for the future analysis but certainly helps to understand better the diffeological framework,especially why it is so useful and efficient. There are no new claims here but all the detailed andelementary proofs we invented ourselves. Such a self-contained and user-friendly exposition of thesubject is nowhere to be found except here. At the end of this section, we briefly introduce the twodifferent approaches to diffeological tangent spaces – the internal and external – and conclude thatour analysis is insensitive for this choice.

Next we introduce the concept of a groupoid, Lie groupoid and Lie algebroid and discuss thegeneralisation to the diffeological setting, which is an interesting open problem. Since we aremainly interested in an application of this framework to the symmetry structure described in [2], andwe discovered that the task of filling this gap exhaustively is not something we could do on the side,we only filled the gaps relevant for our analysis. The idea is, as suggested in the Appendix of [2],to define the diffeological algebroid through the group of bisections with the bracket being givenby the Lie bracket there. An equivalence of this perspective to the standard one was recently provedfor Lie groupoids on compact bases. It seems then a plausible path to take, although it is not clearunder what assumptions such a construction makes sense and leads to something which we wouldlike to call a diffeological algebroid. However, it does make sense for the case we are interested in –the algebroid we get is a vector bundle equipped with a bracket and an anchor, just like for the smoothcase, with the only difference being that the base space is no longer a manifold but a diffeologicalspace. We leave the general case unsolved.

The second part is an application of the partly-developed framework of diffeological algebroidsto the groupoid structure described in [2]. After introducing the concepts and underlying theirimportance, we present the groupoid of [2] and take care of the second point above by puttinga different diffeological structure on the groupoid, which we find simpler and more natural.

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Next, by a detailed analysis of the group of bisections, we confirm the choice of g-gaussian extensionsof vector fields defined on hypersurfaces as representing sections of the algebroid which, althoughcrucial for establishing the bracket structure, we did not find to be done very clearly in the originalapproach [2]. We hope to have thus improved the understanding of the origin of this structure,certainly we clarified it for ourselves. We also provide a new reasoning pointing at the samerepresentation via the Lie algebra structure of the group of bisections. On the way, we may havealso provided an interesting link to the approach that Weinstein and Blohmann are sketching at theend of [3]. There they develop a general theory of Hamiltonian Lie algebroids, still motivated by thedesire of understanding the dynamics of the Einstein’s theory. The diffeological structure of ourchoice also simplifies the global structure of the derived algebroid, the approach we present is stillinvolved, but in our view more straightforward than the original one, which was our third goal. Thetriviality of the algebroid bundle, mentioned but not proved in the original paper, is easily seen fromthe point of view of our analysis.

We close by summarising what has been done and sketching some interesting lines of furtherdevelopment of this approach. Among other perspectives, we describe the original motivation of Prof.Klaas Landsman for looking into these structures – namely, to see if it could be linked to a quantizationframework based on integration of Lie algebroids that he developed in [12].

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Part I

Mathematical frameworkIn our view, the main purpose of Mathematical Physics is to provide the appropriate mathematicalframeworks for capturing theories of Physics in order to allow for better understanding of the theoriesthemselves and, ultimately, the physical reality they describe. We think of a framework as appropriateiff it is capable to grasp all the physical notions that appear in the theory and allows for unambiguousand precise reasoning with them.

However, what we are going to do here does not fall exactly in the scope of this definition – we aimto provide an appropriate mathematical framework for providing a partial answer to a meta questionabout the structure of a physical theory, namely the theory of General Relativity. The question canbe phrased as follows:

What is the origin of the bracket structure of the constraints of General Relativity?

The partial answer that we want to support and improve was given in [2] where the authors putforward a claim that an identical structure is given by the constant sections of an algebroid describingdiffeomorphism invariant deformations of space-like hypersurfaces in Lorentzian manifolds, whichcan be derived from a groupoid that captures the global symmetry structure of the initial valueformulation of the Einstein equation.

As we will see, a mathematical framework powerful enough to not only encode this symmetrystructure, but also to support the analysis of the associated infinitesimal structure, i.e. appropriate forthe task, is that of diffeological algebroids. In this part we aim to introduce all the necessary conceptsand provide a sufficient technical and conceptual basis.

We begin by providing basics of the theory of diffeological spaces, which is one of the generalizationsof the manifold framework. The advantages of diffeological spaces over manifolds are, among others,the simplicity of treatment of functional and quotient spaces – we will need both those features for thesmooth description of our groupoid. Next we dive into the analysis of the structural properties of thetheory of diffeological spaces, providing a detailed, self-contained and user-friendly description ofthe sheaf-theoretic perspective on this framework, which is not directly needed for the analysis butdeepens the understanding of this tool.

In the second part we present the concept of a groupoid, Lie groupoid, and its associated Liealgebroid, and push it a bit in the direction of diffeological algebroids, establishing the ground forour future analysis. We discuss some perspectives and difficulties of developing a general notionof a diffeological Lie algebroid, which is central to our task and to the best of our knowledge has notbeen treated in the literature before.

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1 Diffeological spaces

Diffeology, next to the theory of (very similar) Chen spaces, Frölicher spaces, Differential spacesof Sikorski etc., is an extension of differential geometry. It was introduced by Jean-Marie Souriauin 1979 and developed in a textbook [10] by Patric Iglesias-Zemmour. More recently, topologyof diffeological spaces and different approaches to diffeological tangent spaces were developed andsummarized in the papers [6] and [5] by J.D. Christensen, E. Wu and G. Sinnamon. As nicely putby Patrick [10]:

With a minimal set of axioms, diffeology allows the geometer to deal simplybut rigorously with objects which do not fall within the usual field of differentialgeometry: quotients of manifolds (even non-Hausdorff), spaces of functions, groupsof diffeomorphisms, etc. The category of diffeological spaces is stable understandard set-theoretic operations, such as quotients, products, coproducts, subsets,limits, and colimits. With its right balance between rigor and simplicity, diffeologycan be a good framework for many problems that appear in various areas of physics.

Sharing his belief in a huge potential for applications of this beautiful concept in different areasof mathematical physics, we aim to provide a relatively self-contained and accessible descriptionby introducing all the notions from the ground up and providing detailed and elementary proofs thatcan not be found elsewhere.

First, we introduce the concept of a diffeological space as sets equipped with a specified set of smoothparametrizations (plots) subject to the three natural conditions (smooth compatibility, covering andsheaf conditions), describing along the way how manifolds fit into this framework. Next, we presentbasic constructions that allow to naturally equip products, sub-spaces, quotients, functional spacesetc. of diffeological spaces with their own, inherited diffeological structure.

Next, we switch to the the sheaf-theoretic perspective - after reflecting on the already discovered prop-erties of the category {Diffeo} of diffeological spaces and smooth maps between them, we carefullyintroduce the basic notions from the abstract sheaf theory. We then show how the category {Diffeo}fits into this framework by proving its equivalence (Theorem 1.18) to the category cShCov{Eucl}of concrete sheaves on the Euclidean site modelled on the category {Eucl} of open subsets of Eu-clidean spaces and smooth mappings. Having achieved that, we explore how the nice propertiesof the category of diffeological spaces can be understood from this perspective.

Finally, we describe different kinds of tangent structures that can be associated to diffeological spaces– again using the power of the Category Theory, we describe the concepts of an internal and externaltangent spaces and discuss some of their properties. We also introduce the very natural notionof a diffeological infinite jet bundle. Even though it can be argued that Frölicher spaces or Differentialspaces are better suited to deal with the tangent structures – there is less confusion about how theseconcepts should be generalized from the manifold framework – we find the richness of the theoryof diffeological tangent spaces not only interesting but also potentially advantageous.

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1.1 Basics

We will now introduce the basic notions and constructions from the theory of diffeological spaces.

1.1.1 Definitions

In this paragraph, we would like to introduce the notion of a diffeological space and diffeologicallysmooth map and explain how they can be understood as a generalization of a manifolds and smoothmaps between them. Let us begin with the definition of a diffeological space:

Definition 1.1. A parametrization of a set X is a map φ : U → X, where U is an open subsetof a Euclidean space Rn of arbitrary dimension n ∈ N.Definition 1.2. A diffeological space, denoted (X ,DX ) is a pair of sets, where X is the set we areconcerned with and DX is the set of parametrizations of X, called plots, subject to the followingnatural conditions:

i) DX is closed under composition with smooth maps (smooth compatibility condition):

φ : U → X , { f : U ′→U} ∈C∞(U ′,U) ⇒ {φ ◦ f : U ′→ X} ∈ DX ,

ii) compatible plots defined on an open cover of an open subset of a euclidean space canbe (uniquely) glued together to give another plot (sheaf condition):

∀{{φi :Ui→X}∈DX ,U =

⋃i

Ui}

: φi|Ui∩U j = φ j|Ui∩U j ∃!{φ :U→X}∈DX : φ |Ui = φi ∀i,

iii) all constant maps are plots (covering condition):

φ : U 3 u 7→ φ(u) = x ∈ X ∀u∈U ⇒ φ ∈ DX ,

We refer to the set DX as a diffeology or a diffeological structure on X.

Definition 1.3. On any set X we can put one of the “extreme” diffeologies: the one that considersall maps to X as plots, called the coarse diffeology, and the one for which the only plots are constantmaps - the discrete diffeology.

Very much like topology, diffeology can be generated:

Definition 1.4. Let F denote some family of parametrizations of a set X. The diffeologygenerated by F , denoted 〈F 〉, is the finest (smallest) diffeology on X containing F :

〈F 〉 :=⋂

F⊂DX

DX ,

where DX runs over all the diffeologies on X.

Further, a diffeological structure does not require an underlying topology – it generates one:1

Definition 1.5. The D-topology on a diffeological space (X ,DX ) is the finest topology making allthe plots smooth:

A ∈ OD(X) ⇔ φ−1(A) ∈ O(U) ∀ {φ : U → A⊂ X} ∈ DX ,

where OD(X) denotes the set of all subsets of X that are DX -open.

1For the complete treatment of D-topologies we refer to [6].

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Among other perspectives, the notion of a diffeological space can be regarded as generalization of thatof a smooth manifold:

Lemma 1.1. A manifolds becomes a diffeological space when plots are defined to be thoseparametrizations that are smooth (in the manifold sense).

Proof. We will use small roman numbering to refer to the Definition 1.2 of a diffeological space.A map f : Rn ⊃U → M to the m-dimensional manifold M is smooth (in the manifold sense) ifffor any local chart χ : M ⊃ A→V ⊂ Rm, the map χ ◦ f : U →V is a smooth (subsets of) betweenEuclidean spaces. Constant parametrizations composed with charts give rise to constant maps, whichare smooth – this assures (i); composition of smooth maps between open subsets of euclidean spaces(that arise after the composition with charts) is again smooth, which gives compatibility requirement(ii); finally, compatible family of smooth maps fi : Ui→M defined on an open cover

⋃Ui =U gives

rise to a family of smooth maps χ ◦ fi : Ui→V and hence can be uniquely glued to a smooth mapon the whole U and since this needs to work for any local chart, we get a smooth map f : U →M andhence (iii) also holds.

Definition 1.6. We call the diffeology described above the manifold diffeology.

From now on, we consider all manifolds as diffeological spaces equipped with manifold diffeologies.Let us now have a look at the concept of a smooth map between diffeological spaces:

Definition 1.7. We call a function between diffeological spaces f : (X ,DX )→ (Y,DY ) a smooth map,denoted f ∈C∞(X ,Y ), iff it takes plots to plots, i.e. it’s composition with any plot on the source spacegives a plot on the target space:

{ f : X → Y} ∈C∞(X ,Y ) ⇔ { f ◦φ : U → Y} ∈ DY ∀{φ : U → X} ∈ DX .

Remark. Clearly, the identity map is always smooth.

Let us first acknowledge some nice properties of the smooth maps just defined:

Lemma 1.2. Smooth maps between diffeological spaces compose.Proof. Let X Y Z

f gand φ ∈ DX . Since f is smooth, { f ◦φ} ∈ DY . Hence, since g is

also smooth, we have {g ◦ ( f ◦φ) = (g ◦ f )◦φ} ∈ DZ and since φ was arbitrary, we get that g ◦ fis smooth itself.

Lemma 1.3. Smooth maps are D-continuous.Proof. Let f ∈ C∞(X ,Y ), A ∈ OD(Y ). Since f is smooth, for any φ ∈ DX we have f ◦ φ ∈ DY .Further, since A is DY -open, for any φ ∈ DX we have that ( f ◦φ)−1(A) = φ−1( f−1(A)) is open, andhence, since φ was a plot on X , we get that f−1(A) is DX -open.

And back to the manifold example:

Lemma 1.4. Smooth maps between manifolds are precisely smooth in the diffeological sense, whenmanifolds are considered diffeological spaces equipped with manifold diffeological structures.

Proof. Let f : M→N, where M and N are manifolds equipped with manifold diffeologies, and denoteby χ and η the local charts defined on the neighbourhoods of x ∈ M and f (x) ∈ N, respectively.Then, since χ and η were arbitrary, f is smooth as a map between manifolds iff χ−1 ◦ f ◦η is smoothas a map between Euclidean spaces. Now take a plot on M, i.e. a map φ : U →M such that χ ◦φis smooth (again as a map between Euclidean spaces). Moreover, let us assume that f ◦φ(U) fits intothe domain of η . Then f is smooth as a map between diffeological spaces iff for each φ as abovewe have that f ◦φ is a plot on N, and hence η ◦ f ◦φ is a smooth map between Euclidean spaces. Butφ = χ−1 ◦χ ◦φ and hence we can write η ◦ f ◦φ = (η ◦ f ◦χ−1)◦(χ ◦φ). We now see that, that since

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(χ ◦φ) is smooth, η ◦ f ◦φ is smooth iff (η ◦ f ◦χ−1), i.e. the requirements of manifold-smoothnessand diffeological smoothness are indeed equivalent.

From now on, unless stated otherwise, we will use the term ’smooth map’ for maps betweendiffeological spaces (including manifolds) that are smooth in the above sense. We also give C∞(X ,Y )the universal meaning of diffeologially smooth maps.Remark. When we put on the open subsets of euclidean spaces the manifold diffeology, plots on a setX are precisely the smooth maps between diffeological spaces:

φ : U → X ∈ DX ⇔ φ ∈C∞(U,X),

and hence there is no ambiguity in referring to plots as smooth parametrizations.

Two diffeological spaces are indistinguishable as such iff they are diffeomorphic – just like formanifolds, we define:

Definition 1.8. Two diffeological spaces (X ,DX ) and (Y,DY ) will be called diffeomorphic, denoted(X ,DX ) ' (Y,DY ), iff there is smooth bijection with a smooth inverse between them. Such a mapis called a diffeomorphism.

Remark. Clearly, (X ,DX )' (Y,DY ) iff X ∼= Y and DX ∼= Y,DY , where by "∼=" means bijective sets.

1.1.2 Basic constructions

In this paragraph we describe the basic constructions concerning diffeological spaces. In particular,we will explain how natural diffeological structures can be put on subsets, cartesian products, quotientand functional spaces, as well as on coproducts, fibered products and pushout sets. We will also provethe smoothness with respect to this diffeologies of the maps naturally arising in this contexts.

Definition 1.9. The subspace diffeology, denoted DA⊂X , is given on a subset A⊂ X of a diffeologicalspace (X ,DX ) by taking plots to be those parametrizations which composed with the inclusioni : A ↪→ X give plots on X:

{φ : U → A} ∈ DA⊂X ⇔ {i◦φ : U → X} ∈ DX .

The subspace diffeology is simply the coarsest one for which the inclusion map is smooth. Naturally,we have:

Lemma 1.5. The restriction of a smooth map to the subset is again smooth for the subspace diffeology.

Proof. Take a smooth map { f : X → Y} ∈ C∞(X ,Y ) and a subset i : A ↪→ X . The restricted mapf eA : A→ Y is smooth for the subspace diffeology iff for any plot φ ∈ DA⊂X , we have f eA ◦φ ∈ DY .Now, since f ∈C∞(X ,Y ) and i◦φ ∈ DX , we have { f ◦ (i◦φ) = ( f ◦ i)◦φ = f eA ◦φ} ∈ DY .

Definition 1.10. The product diffeology DX×Y is given on a cartesian product X×Y of diffeologicalspaces (X ,DX ) and (Y,DY ) by taking plots to be those parametrizations which, when composed withprojections, are plots on the factors:

φ : U → X×Y ∈ DX×Y ⇔ πX ◦φ ∈ DX & πY ◦φ ∈ DY ,

where πX and πY are the canonical projections.

We see that the product diffeology is in turn the finest one making the projections πX and πY smooth,and that the definition can be easily generalized to arbitrary finite products. Somewhat similarly,we have a natural diffeological structure on a disjoint union:

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Definition 1.11. The quotient diffeology is a diffeology Dπ(X) given on an image π(X) of a surjectivefunction π defined on a diffeological space (X ,DX ) by taking plots to be those parametrizations thatare given by composition of π with a plot on X:

φ : U → π(X) ∈ Dπ(X) ⇔ φ = π ◦ψ, ψ ∈ DX .

This is the coarsest diffeology making π smooth. Given an equivalence relation on a diffeologicalspace, the canonical projection π : X → X/∼ provides the diffeology on the quotient space, whichthe given name.

We can also easily define a diffeological structure on the space of smooth functions between diffeo-logical spaces:

Definition 1.12. The functional diffeology is a diffeology DC∞(X ,Y ) given on a set of smooth func-tions C∞(X ,Y ) between two diffeological spaces (X ,DX ) and (Y,DY ) by taking plots to be thoseparametrizations for which the evaluation map is smooth:

φ : U →C∞(X ,Y ) ∈ DC∞(X ,Y ) ⇔ evφ : U×X → Y ∈C∞(U×X ,Y ),

where evφ : U ×X 3 (u,x) 7→ φ(u)(x) ∈ Y is the evaluation map and we consider the manifolddiffeology on U and the product diffeology on U×X.

As an illustration of the introduced notions, let us prove the following simple fact in details:

Lemma 1.6. The composition of smooth maps is a smooth map between functional spaces.

Proof. Consider { f : X → Y} ∈ C∞(X ,Y ) and {g : Y → Z} ∈ C∞(Y,Z) and equip the functionalspaces with the functional diffeologies just defined. The composition of smooth functions is a map:

C∞(X ,Y )×C∞(Y,Z) 3 ( f ,g) 7→ cmp( f ,g) := g◦ f ∈C∞(X ,Z),

and we consider the product-functional diffeology on the domain and the functional diffeology on thecodomain. It is smooth iff for any plot φ : U 3 u 7→ ( fu,gu) ∈C∞(X ,Y )×C∞(Y,Z) the map:

cmp◦φ : U 3 u 7→ gu ◦ fu ∈C∞(X ,Z)

is a plot on C∞(X ,Z). Now, φ is a plot on a product space, i.e. we have that both maps:

π1 ◦φ : U 3 u 7→ fu ∈C∞(X ,Y ) & π2 ◦φ : U 3 u 7→ gu ∈C∞(Y,Z)

are smooth, which because of the functional diffeology that we put on C∞(X ,Y ) and C∞(Y,Z)translates to the smoothness of the relevant evaluation maps:

ev{π1◦φ} : U×X 3 (u,x) 7→ fu(x) ∈ Y & ev{π2◦φ} : U×Y 3 (u,y) 7→ gu(y) ∈ Z.

Hence, since the composition of smooth maps is again smooth (Lemma 1.2), we get that the map:

ev{cmp◦φ} : U×X 3 (u,x) 7→ gu( fu(x)) ∈ Z

is smooth, which means precisely that cmp◦φ is a plot on C∞(X ,Z).

For the sake of completeness, let us also spell out how disjoint unions, fibered products and pushoutsets are also naturally equipped with inherited diffeological structures:

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Definition 1.13. The coproduct diffeology DXtY is given on a disjoint union X tY (coproduct)of diffeological spaces (X ,DX ) and (Y,DY ) by taking the plots to be those parametrizations thatlocally factor through a plot in either X or Y , i.e. for every connected component Ui ⊂U of an opensubset of a Euclidean space we have:

{φ : U → X tY} ∈ DXtY ⇔ {∃ψi∈DX : φ∣∣Ui

= iX ◦ψi} ∨ {∃ηi∈DY : φ∣∣Ui

= iY ◦ηi},

where iX : X → X tY ← Y : iY are the canonical inclusions.

This is the coarsest diffeology making the inclusions iX and iY smooth, and we see that plotsof coproduct diffeology on a disjoint union space are locally given along factors and that thedefinition above can also be easily generalized to arbitrary finite coproducts. Let us first recall thedefinitions of a fibered product and a pushout set:

Definition 1.14. Given two functions with the same codomain, f : X → Z and g : Y → Z, we definethe fibered product of X and Y via:

X×Z Y := {(x,y) ∈ X×Y : f (x) = g(y)} ⊆ X×Y.

A pushout set is a notion similar in some sense:1

Definition 1.15. Given two functions with the same domain, f : Z→ Z and g : Z→ Y , we define thepushout set to be:

X tZ Y := X tY/∼,

where “∼” denotes the following equivalence relation: x∼ y iff ∃z∈Z such that f (z) = x and g(z) = y.

Combining the constructions already introduced, we can easily put diffeological structures on them:

Definition 1.16. We equip a fibered product set X×Z Y with the subspace-product diffeology, denotedDX×ZY . This diffeological structure is referred to as the pullback diffeology.

Definition 1.17. We equip a pushout set with the coproduct-quotient diffeology, denoted DXtZY , andrefer to it as the pushout diffeology.

1We will make this ’sense’ precise in the next section, after introducing limits and colimits.

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1.2 Sheaf-theoretic perspective

The category {Diffeo} of diffeological spaces and smooth maps is equivalent to the category of con-crete sheaves on the Euclidean site (Theorem 1.18). We find this perspective not only very appealingbut also enlightening. The language of sheaf theory we are going to introduce, allows to see theconcept of a diffeological space in a different light. Especially, the nice properties of {Diffeo}that we have seen can be more deeply understood when we adapt this perspective. After carefullyintroducing general concepts, we aim to simplify things by making them explicit, providing detailed,elementary proofs and always looking for an optimal way of presentation. Our purpose is, exceptdeepening our own understanding of this beautiful concept, to produce a relatively self-contained andcomprehensible presentation of the theory of diffeological spaces with some didactic flavour. We referto [1] for a slightly more complete treatment and to the relevant articles on nlab that we found veryuseful while writing this section.

1.2.1 Categorical preliminaries

While introducing basics of the Category Theory, we will also provide some important examplesto be used later on and establish some notation. Let us start from the very beginning:

Definition 1.18. A category C = {C0,C1} is a basic mathematical structure that consists of a setof objects C0 and a set1 of arrows C1 connecting them, subject to the following conditions:

i) arrows that meet can be composed (closedness of composition):

∀ f ,g ∈ C1 : A B Cf g ∃(g◦ f ) ∈ C1 : A B Cf

g◦ f

g commute,

ii) order of composition does not matter (associativity of composition):

f ◦ (g◦h) = ( f ◦g)◦h,

iii) for each object C ∈ C there is an identity arrow IdC, pointing from C to itself, such that:

∀ { A Cf } : IdC ◦ f = f , & ∀ { C Bg } : g◦ IdC = g.

Remark. We will use the terms arrow and morphism interchangeably. By a diagram in C we meana drawing with objects of C represented (usually) by capital Latin letters and arrows representingmorphisms between them, usually ’named’ with the use of small Latin letters. We also introducethe notation C (A,B) for the collection of all morphisms in C1 from A ∈ C0 to B ∈ C0. In CategoryTheory, we consider two arrows f , f ′ ∈ C (A,B) to be the same, denoted f = f ′, iff the diagram:

A Bf ′

f

commutes, i.e the arrows f and f ′ can be interchanged in any diagram in C with no effect, and thusare indistinguishable in the realm of the category C . Inherent in the category structure of C are twoprojections s, t : C1→ C0 called the source and target projections, that take an arrow to it’s startingor ending point, respectively:

C0 3 A {A B} B ∈ C0,fs t

e.g. two arrows, f and g, can be composed to (g◦ f ) iff t( f ) = s(g).

1In general, neither C0 nor C1 are assumed to be sets, but merely proper classes. However, we do not needsuch a level of generality for our purposes. A category as we define it is usually called small.

12

Many interesting categories consist of objects that are sets, or sets with some additional structure,1

and arrows that are just maps preserving this structure and forming a set that is closed with respectto the (associative) composition and contain the identity functions. Let us here give some examplesof those that we will be referring to later on and introduce some notation:

• {Set} denotes the category of sets and functions,• {Vect} denotes the category of vector spaces and linear maps,• {Mfld} denotes the category of manifolds and smooth maps,• {Diffeo} denotes the category of diffeological spaces and smooth maps.

As we have seen in the previous section, given a pair of diffeological spaces we can form a third onewith the cartesian product or a disjoint union as the underlying set with a relevant, canonical diffeo-logical structure, and that this construction generalizes to arbitrary finite products and coproducts.We say that the category {Diffeo} has finite products and coproducts.

We have also seen that the set of smooth functions between any two diffeological spaces is alsonaturally equipped with a canonical diffeological structure – the functional diffeology. We say that{Diffeo} is closed. The diffeological space C∞(X ,Y ) is called the exponential object of (X ,DX ) and(Y,DY ). A category that is closed and has finite products, e.g. {Diffeo}, is called cartesian closed.

Last but not least, we have shown that a diffeological structure is naturally inherited from a diffeolog-ical space on any quotient space.

Notice also that there is a distinguished one-point diffeological space – indeed, there is only one wayin which we can equip the one-point set with the diffeological structure, namely by declaring allof the functions from the open subsets of Euclidean spaces into {∗} to be plots. Notice, that in thisdegenerate case the coarse and discrete diffeologies coincide. This diffeological space has a propertythat for any other one (X ,DX ) the is a unique function !X : X →{∗}, which is trivially smooth sinceevery parametrization into {∗} is a plot. Such an object in a category – with a unique arrow from anyother object – is called a terminal object.

We also have another distinguished diffeological space with a similar property – the one modelledon the empty set. Indeed, since there are no maps into the empty set /0, there is only one way we canmake it a diffeological space, namely by declaring D /0 := /0, i.e. there are no plots on /0. All the axiomsare then trivially satisfied, and since there is always a unique function from /0 to any diffeologicalspace – the empty function, since there are no elements in /0 – which is trivially smooth since thereare no plots on /0. Such an object in a category – with a unique arrow to any other object – is calledan initial object.

The category {Mfld} deals relatively well with products – we have a natural manifold structureon cartesian products of manifolds – but it fails to simply include the other derived spaces:

(a) The disjoint union of two manifolds M1tM2 is naturally a manifold itself iff they have thesame dimension: dim(M1) = dim(M2).

(b) The theory of infinite-dimensional manifolds is rich and difficult, it’s often far from obviousif a given functional space is a manifold or not (and in which sens this question shouldbe conveniently considered) [citation].

(c) The quotient space M/G, where M is a manifold and G a Lie group is a manifold iff theaction of G on M is free and proper [citation].

Thus we see that the category {Diffeo} has many structural advantages over the category {Mfld}.Indeed - the simplicity in which it deals with functional spaces and quotients is precisely the reasonfor which it will be a convenient framework for our purpose.

1If such a category admits an object formed on the one-point set, we call it concrete.

13

As a next step towards the theory of sheaves we introduce the concept of a functor, which canbe understood as a map between two categories respecting their structures. But before doing so, letus introduce a few more example of categories that will enrich our exposition and will be of future use:Example 1.7. A group (G, ·) can be understood as a category G , called a group category, with oneobject, i.e. G0 = {∗} and arrows from it to itself corresponding to each if the group elements. Theidentity arrow Id∗ corresponds to the group identity e ∈G, associativity and closure conditions beingalready present in the definition of a category. Notice also, that because of the group structure eacharrow has a two-sided inverse. Groups are then one-point categories with all the arrows invertible.Example 1.8. By {Eucl} we will denote the Euclidean category with open subsets of Euclideanspaces as objects and smooth maps between them as arrows. It can be seen as a full sub-categoryof the category {Mfld}, meaning that its objects are manifolds and morphisms are precisely thesmooth maps between them.Example 1.9. Any topological space (B,T ) defines a category, denoted by B, with open subsetsof B as objects: B0 =O(B), and arrows given by inclusions: U→V ∈B1 iff U ⊆V . Closedness forcomposition follows since the inclusion is transitive, associativity is obvious and we have identitiessince U ⊆U always.Definition 1.19. Given two categories C and D a functor F : C →D , is a pair of parallel maps:

F = {F0 : C0→D0, F1 : C1→D1},

such that the image of any diagram in C is a diagram in D , i.e. F:

i) respects the source and target projections: s(F1( f )) = F0(s( f )) & t(F1( f )) = F0(t( f )),

ii) is compatible with the composition: F1(g◦ f ) = F1(g)◦F1( f ),

where by the abuse of notation we denote by s, t and ”◦ ” the source projection, target projection andcomposition, respectively, of both categories.Remark. The first requirement means that for all A,B ∈ C0 and any f ∈ C (A,B) we have:

F : {A B}f 7→ {F0(A) F0(B)}.F1( f )

Remark. It follows that F respects the identity morphisms: F1(IdC) = IdF0(C).Example 1.10. We always have the identity functor, which sends each object and arrow to itself.Example 1.11. We can think of a functor that sends sets with some additional structure to simplesets and special maps preserving the given structure to themselves, e.g. U : {Vect} → {Set}. Thisis an example of a forgetful functor, i.e. one that ’forgets’ part of the structure.Example 1.12. A group homomorphism h : (G, ·)→ (H,∗) is the same thing as a functor G →H .Indeed, the requirement h(g ·g′) = h(g)∗h(g′) for g,g′ ∈ G can as well be understood as the secondcondition on the arrows that represent g and g′ in G , first one being trivial since both the categorieshave only one object. This is a perfect example of a functor that simply respects the categoricalstructures.Example 1.13. We have a functor {Eucl} ↪→ {Mfld} that reinterprets open subsets of Euclideanspaces as special kind of manifolds. Similarly, we have a functor {Mfld} ↪→ {Diffeo}, whichunderstands manifolds as special diffeological spaces. This are examples of inclusion functors.

A functor as defined above is sometimes called covariant to distinguish between another class of verynatural objects:Definition 1.20. Given two categories C and D a contravariant functor F : C → D , is a pairof parallel maps:

F = {F0 : C0→D0, F1 : C1→D1},such that the image of any diagram in C is a diagram in D with reversed arrows, i.e. F:

14

i) switches the source and target projections: s(F1( f )) = F0(t( f )) & t(F1( f )) = F0(s( f )),

ii) is compatible with the composition (of reversed arrows): F1(g◦ f ) = F1( f )◦F1(g).Remark. The first requirement now means that for all A,B ∈ C0 and any f ∈ C (A,B) we have:

F : {A B}f 7→ {F0(A) F0(B)}.F1( f )

Remark. A contravariant functor C →D is the same thing as a covariant functor C op→D , whereC op denotes the category constructed from C by reversing the direction of all of the arrows, whichis the same thing as switching s for t and vice versa.

1.2.2 Sheaves on sites

We are now ready to present the archetypal sheaf that will serve us an illustration and motivationfor the more general concept that we are going to develop in this paragraph. Consider a sheafof continuous functions on a topological space:

Example 1.14. Given a topological space (B,T ) we define a contravariant functor F : B→{Set},or equivalently a functor C op → {Set}, called a sheaf of continuous functions on B, that assignsto any open subset U ∈O(B) the set F(U) of continuous real-valued functions defined on U, and therestriction map eV,U : F(V ) 3 f 7→ f |U ∈ F(U) to each of the arrows U →V in B, i.e. inclusionsU ⊆V . We see, that reversing the arrows is the most natural thing to do in this kind of a situation.It can be pictured as a diagram:

U V

F(U) F(V )

⊆

F FeV,U

For the second requirement, let us take U ⊆ V ⊆ W and F ∈ F(W ) = C∞(W,R). Composi-tion of this chain of inclusions is simply the inclusion U ⊆W, so F takes it to the restrictioneW,U : F(W ) → F(U). If we apply the functor to the inclusions separately and compose afterwardsinstead, we get a composition of restrictions eW,V◦eV,U , which is the same thing since ( f |V )|U = f |U .

Let us now point out an important property of the sheaf defined above. Consider an open coverU =

⋃i Ui of some U ∈ O(B) together with a family of continuous functions fi ∈ F(Ui) that are

compatible on the intersections – such a collection will be called a compatible family:

Definition 1.21. Given an open subset U ∈ O(B) with an open cover U =⋃

i Ui we call a familyof continuous functions F = { fi : Ui→ R | i ∈ I} a compatible family iff:

fi|Ui∩U j = f j|Ui∩U j , ∀i, j.

It is an elementary property of continuous functions, that such a compatible family can be gluedtogether to give a continuous function on the whole U . We will refer to this as the gluing propertyand call such a glued function f ∈ F(U) an amalgamation of the compatible family:Definition 1.22. Given an open subset U ∈ O(B) with an open cover U =

⋃i Ui and a compatible

family F = { fi : Ui→ R | i ∈ I} we call the function f ∈ F(U) an amalgamation of F iff:

f |Ui = fi ∀i.

Moreover, the gluing has a unique outcome: given two functions f , f ′ ∈ F(U), we know that if theyagree on each of the members of the covering, they need to agree on the whole U . The elementsof F(U) are then defined locally with respect to the topology on B - this is called the localityproperty. Together, they give that the amalgamations as above not only exist but are unique:

15

For every covering U =⋃

i Ui and a compatible family F = { fi ∈ F(Ui) | i ∈ I} there is a uniqueamalgamation f ∈ F(U) of F .

We will refer to this as the sheaf property. It can be understood as follows: elements of F(U) forthe sheaf F : B→{Set} on any object U ∈B0 is fully determined by their compatible values on anycovering of U .

The functor just described can be generalized in different directions. Firstly, we could considerfunctions to be only k-times differentiable or merely continuous, and so on. As a next step, we couldgive up on thinking of elements of F(U) as functions at all and consider them as abstract elementsof sets that are assigned to open subsets of B instead. However, we would probably like to put someextra conditions on such a generalized object to assure that the nice property described above stillhold, in a more abstract sense. Indeed, this will be the defining property distinguishing sheafs frommore general class of objects called pre-sheafs. Finally, we can also try to generalize this conceptso that it makes sense for a broader class of categories than just those that come from topologicalspaces. This is the plan, so bear with us! More concretely, we are going to:

1) Define a pre-sheaf on a general category, forgetting about topological spaces and functions

2) Generalize the notion of an open covering to a general category by introducing sites1

3) Define sheaves in this general setting

4) Show how a diffeological space can be understood as a concrete sheaf on a site of {Eucl}5) State and prove the equivalence of categories {Diffeo} and the category of those sheaves

Definition 1.23. A pre-sheaf F on a category C is a functor from the opposite category to {Set}, i.e.:

F : C op→{Set}.

Simple as that. Notice here, that we have a natural notion of sub-presheaf. Indeed, given a pre-sheafF : C op→{Set} we can think of a sheaf H which sends objects of C to the subsets of the sets thatare assigned by F and respects the arrows:

Definition 1.24. A sub-presheaf H of F : C op→{Set}, denoted by H ⊆ F, is a pre-sheaf on C sat-isfying:

H(C)⊆ F(C) ∀C∈Co , H( f ) = F( f )∣∣H(C) ∀ f∈C (A,C).

Now, all we need to do in order to state the sheaf property in the more general context is to generalizethe notions of an open cover, a compatible family and an amalgamation.

There are various ways to generalize a notion of an open cover to an arbitrary category, theGrothendieck topology being probably the most classical and widely used one. However, we find thenotion of a coverage [16] advantageous for the following reasons:

1. it is simpler and easier to introduce – there is only one condition that we need to imposeon the coverings and no additional notions,2

2. it is more natural as a generalization of the topological notion – taking the standard opencovers constitute a coverage for B, which is not the case for the Grothendieck topology,3

1We actually do not need the full generality as on out category the notion of an open cover is very naturaland we feel that does not necessary need to be that strongly supported. Hence, the reader not interested in thegeneral theory of sheaves (what a pity!) is welcome to skip this part and move directly to the Definition 1.29.

2Grothendieck distinguishes covering sieves and imposes three conditions in his definition. See also [14]and [11].

3Sieves are by definition downwards-closed while topological coverings very well might not be.

16

and yet it gives rise to a Grothendieck topology and moreover defines the same sheaves [1]. We willthen use the concept of a coverage to define sites, and later sheaves.

Let us denote the collection of open covers of U by Cov(U). In terms of sets, we write U =⋃

i Ui andrequire all the Ui’s to be open. However, if we want to free ourselves from the context of topologicalspace, we need to focus on the properties that are natural in the categorical context, i.e. on the arrows,and forget about set-theoretic operations. Notice first that since an open cover of U is a familyof open subsets of U , for each Ui we have an arrow Ui →U in B. As a property distinguishinga covering SU = {Ui ∈ O(B) | i ∈ I,

⋃i Ui = U} ∈ Cov(U) from an arbitrary collection of open

subsets of U we can choose the following: for any open subset V ⊆U , we have the intersectionSU ∩V = {Ui∩V | i ∈ I} that covers V . In other words, for any SU ∈Cov(U) and V ⊆U we haveSU ∩V ∈ Cov(V ). Clearly, since U is open itself, if the above holds it follows that SU coversU . Moreover, for each element Ui∩V of the covering SU ∩V we have a natural inclusion Ui∩V ⊆Ui.The idea is now to reformulate this property in terms of arrows in B1 alone, i.e. without referringto the set-theoretic intersection, and treat it as an essential characteristic of the notion of an opencover. Writing g∗(SU ) for SU ∩V and Vi for Ui ∩V , the above property may be then formulatedin abstract terms as follows:

For any covering SU ∈ Cov(U) and a subset g : V ⊆ U there exist a covering g∗(SU ) ∈ Cov(V )such that for all of its members {iV : Vi ⊆V} ∈ g∗(SU ) we have a member of the original covering

{i : Ui ⊆U} ∈ SU and an inclusion fi : Vi ⊆Ui such that g◦ iv = i◦ fi, i.e we have1:Vi Ui

V U

iV

fi

ig

.

We call g∗(SU ) a pull-back of SU along g and refer the above property as stability under pull-back. Now, equipping an arbitrary category with a coverage is nothing but declaring which of thefamilies of morphisms we consider to be covering. Since the characterization above generalizesstraightforwardly to an arbitrary category [16], we can simply define:

Definition 1.25. A coverage on a category C , denoted by Cov, is a pull-back stable assignmentof a collection Cov(C) of families of morphisms into C to each of the objects C ∈ C0, i.e. we have:

∀{SC ∈Cov(C), g : A→C} ∃ g∗(SC)∈Cov(A) : ∀ j∈ g∗(SU ) ∃ {i j ∈ SC, f j ∈C1} :A j Ci

A C

j

f j

i j

g

.

Elements of Cov(C) are called coverings of C, a pair (C ,Cov) is called a site.

Remark. The category B is naturally a site if we declare coverings to be the usual open covers.

Now, when we know what means for a family of morphisms in a site to be covering, we are almostready to state the condition for a presheaf on (C ,Cov) to be a sheaf. But first, because we decidedto forget about functions, we need to state what means for a family of elements xi ∈ F(Ci), whereF is a pre-sheaf on C and Ci ∈ C0 for all i, to be compatible. Again, we need to get rid of a set-theoretic operation of intersecting elements of an open cover that appear in the definition of a coveringfamily in the context of B. The idea here is to exchange the requirement that the elements of thefamily agree on all intersections of the members of a covering for the equivalent one, namely thatthey agree on all of their common subsets:

Definition 1.26. Given a pre-sheaf F : C →{Set} on a site (C ,J), a compatible family for a covering{i :Ci→C | i∈ I}∈Cov(C) is a collection of elements xi ∈F(Ci), such that for any pair of morphismsai : A→Ci and a j : A→C j with i◦ai = j ◦a j we have F1(ai)(xi) = F1(a j)(x j) ∈ F(A).

1We believe that only the commutative diagrams are worth drawing, and hence do not indicate the commuta-tivity in any way.

17

Remark. The pair of morphisms in the above definition corresponds to a subset of an intersectionof Ui∩U j, while F1(ai) and F1(a j) are the relevant restrictions.

Just like for a sheaf of continuous functions, we define an amalgamation of a compatible family:

Definition 1.27. Given a pre-sheaf F : C → {Set} on a site (C ,Cov) and a compatible familyxi ∈ F(Ci) for a covering {i : Ci→C | i ∈ I} ∈Cov(C), is an amalgamation of xi ∈ F(Ci) an elementx ∈ F(C) such that F1(i)(x) = xi for all i.

Finally, we define sheaves on (C ,Cov) to be those pre-sheaves that satisfy the sheaf property:

Definition 1.28. A pre-sheaf F : C op→{Set} on a site (C ,Cov) is called a sheaf iff it satisfies thesheaf property, i.e. for any covering {i : Ci→C | i ∈ I} ∈Cov(C) and a compatible family xi ∈ F(Ci),there always exists a unique amalgamation.

1.2.3 Category of diffeological spaces as a sheaf category

We are really close now to being able to see that a diffeological space is actually a sheaf on a site,and how the smooth functions can be understood from this point of view, which is the focus of thisparagraph.

The site we are interested in is formed on {Eucl} – the category of the domains of the plots, whichis in a way very close to being a topological space: each object U ∈ {Eucl}0 is an open subsetof a Euclidean space, U ⊆ Rn for some n ∈ N, and hence admits a natural notion of an open covercoming from the topology on Rn:

Definition 1.29. Since the inclusions are smooth, we can define a coverage on {Eucl}, end hence theEuclidean site, by simply declaring the coverings for Rn ⊇U ∈ {Eucl}0 to consist of the topologicalopen covers of U in Rn.

Lemma 1.15. The coverage on {Eucl}, as just defined, satisfies the pull-back stability requirement.

Proof. Let’s fix U,U ′ ∈ O(Rn), an open cover {Ui | i ∈ I} ∈ Cov(U), i.e. a collection of opensubsets of U such that U =

⋃i Ui, and a smooth function g : U ′→U . We can define the pull-back

of {Ui | i ∈ I} along g to be:

g∗({Ui | i ∈ I}) := {g−1(Ui) | i ∈ I}.

They are open since g is smooth and hence continuous and Ui’s are open. They cover U ′ because it’sa domain of g and Ui’s cover the whole U . The maps j and fi from the Definition 1.25 are constructedas follows: fi := g|g−1(Ui) and j is the inclusion g

−1(Ui)⊆U ′.

A diffeological space is defined as a set X equipped with some additional structure. If it is aboutto correspond to a sheaf, the set itself has to be a part of its definition. Notice first, that the objectsof {Eucl} have an underlying structure of sets – they are subsets of Euclidean spaces. Moreover,a one-point space {∗} = R0 is clopen and hence an object of the the category1 {Eucl} – it is itsterminal object. We call such a category – with objects formed on sets and admitting a terminal objecton the one-point set – concrete2. For any concrete category C and a fixed set X , we have a verynatural3 pre-sheaf on C which assigns to each object the set of all functions from its underlying setto X and the pre-composition operation to the morphisms:

1This is compatible with the convention that the one-point space is a manifolds.2There is some ambiguity in defining the notion of a concrete category, a concrete site and a concrete

pre-sheaf – namely in the place where we put the requirements connected to the terminal object. However,it does not matter for the notion of a concrete sheaf, which is the object we are really interested in.

3It is almost the same notion as that of a contravariant Hom functor, the difference being that the setsof arrows are taken in {Set}, and not as usually in the category itself.

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FX : C op→{Set}C 7→ {Set}(C,X),

{g : C′→C} 7→{

_◦g : {Set}(C,X)→{Set}(C′,X)}.

We already see that when applied to {Eucl} this gives all functions from open subsets to our chosenset, i.e. all possible parametrizations. Lets then take a look at something smaller:

Definition 1.30. Given a set X, a concrete pre-sheaf DX : C op → {Set} on a concrete categoryC is a sub-presheaf DX ⊆ FX that agrees on the terminal object C to X, i.e. DX ({∗})∼= FX ({∗})1.Remark. Notice that, since points of X correspond one-to-one to functions pointing at them from theone-point set, we have FX{∗} ∼= X . The compatibility on the one-point-set requirement is imposedin order to be able to extract the set X from a concrete functor via X ∼= DX{∗}.

And finally we can define:

Definition 1.31. A concrete sheaf on the Euclidean site will be called a diffeological sheaf.

And appreciate our first important result [1], [15]:

Lemma 1.16. Diffeological spaces are in one-to-one correspondence with diffeological sheaves.

Proof. As noted above, the idea is that DX (U) simply gives all the plots U →DX{∗} ∼= X , and henceDX defines the whole diffeological structure on X – we will now refer to the elements of DX (U), forany U ∈ {Eucl}0, as plots. We divide the proof into the following steps:2

1) Smooth compatibility (i) is equivalent to DX being a sub-presheaf of FX : {Eucl}op→{Set}.

For any smooth f : U ′→U we have DX ( f ) : DX (U)→ DX (U ′) that maps φ 7→ f ◦φ , andhence f ◦φ ∈ DX (U ′), i.e. for any plot φ : U → X the map f ◦φ is a plot U ′→ X if onlyf is smooth with respect to the diffeology on (X ,DX ).

2) The sheaf condition (ii) is equivalent to the sheaf property of DX .

Take an arbitrary covering U =⋃

i Ui, i : Ui ⊆U and a family of plots {φi : Ui→ X | i ∈ I}compatible in the sheaf sense, i.e for all pairs of smooth maps ai : V →Ui and a j : V →U jsuch that i◦ai = j ◦a j we have:

DX (ai)(φi) = φi ◦ai = φ j ◦a j = DX (a j)(φ j).

The sheaf condition on DX says that we then have a unique amalgamation, i.e. a plotφ ∈ DX (U) such that:

DX (i) = φi ∀i∈I .

We will show that this is exactly the same statement as the sheaf condition (ii) by comparingthe relevant definitions.

(a) (coverings) Because of the natural coverage that we put on {Eucl}, the notions of a cov-ering in {Eucl} and the one we standard one we use in (ii) are exactly the same.

(b) (compatible families) The condition i ◦ ai = j ◦ a j gives us that the functionsai and a j have a common image V ′ := ai(V ) = a j(V ) contained in the intersectionV ′ ⊆ Ui ∩ U j, and that they agree on it, i.e. as functions into V ′ they are the same

1By ∼= we denote here the bijection of sets.2The small Roman numbering refers to the Definition 1.2 of a diffeological space.

19

– let us denote the surjective function that they both represent by a : V → V ′. Thesheaf-theoretic compatibility condition then reads:

DX (ai)(φi) = φi ◦a = φi|V ′ ◦a = φ j|V ′ ◦a = φ j ◦a = DX (a j)(φ j),

and since a is an arbitrary map with an image V ′ ⊆Ui∩U j, this means that the plotsφi and φ j agree on all of the common subsets of Ui and U j:

φi|V = φ j|V ∀V ⊆Ui∩U j,

which is equivalent to the compatibility on in the usual sense: φi|Ui∩U j = φ j|Ui∩U j .(c) (amalgamations) We have DX (i) = DX (Ui ⊆ U)(φ) =eUiφ = φ |Ui , and hence the

properties of being an amalgamation also agree in both languages.

We then see that all of the notions that appear in the formulations of a sheaf property coincide,and since statements have exactly the same form the sheaf properties are indeed equivalent.

3) The covering condition (iii) is equivalent to the requirement DX ({∗}) = FX ({∗}) (plus thesmooth compatibility property (i)).

The set DX{∗} consists of plots {∗} → X , which need to be constant. Moreover, anyconstant parametrization (uniquely) factors through {∗}, and hence, because of the smoothcompatibility condition, the set DX{∗} is isomorphic to the subset of X for which constantmaps are plots. Then, requiring DX ({∗}) ∼= FX ({∗}) means that all the constant mapsare plots.

We now see that the concept of a diffeological sheaf is precisely what we want if the generalizedsmooth space that it represents is to be modelled on a set (concrete) with the smooth structurecompatible with the change of parametrizations (pre-sheaf condition) and the topological structure(sheaf condition) of Euclidean spaces.

The collection of diffeological sheaves is naturally equipped with the structure of a category, thearrows being given by natural transformations. Let us define:

Definition 1.32. Given two functors F,H : C → D , a natural transformation, denoted η : F ⇒ His an assignment to any object C ∈ C0 an arrow {ηC : F(C)→ H(C)} ∈ D1 such that for any{ f : C→C′} ∈ C1 we have the (commuting) naturality square:

F0(C) F0(C′)

H0(C) H0(C′)

F1( f )

ηC ηC′

H1( f )

.

It is an easy exercise to check that taking the functors C → D as objects and the natural trans-formations as arrows we get a category structure – composition is composition of the naturalitysquares, associativity follows from their commutativity and the identity functors serve as the iden-tities – we will refer to such categories as functor categories. We then have a category structureon the collection of pre-sheaves on a category C , denoted PSh(C ) and its full subcategory ShCov(C )of sheaves.

Since diffeological spaces are special kinds of sheaves, we can consider the category of diffeologicalsheaves with the arrows given by natural transformations. More concretely, given two diffeologicalsheaves DX ,DY : {Eucl}→ {Set}, a natural transformation between them is an assignment of plots

20

U → Y to plots U → X . Indeed, η : DX ⇒ DY is an assignment to any object U ∈ {Eucl}0 a mapηU : DX (U) ⇒ DY (U) such that for any smooth map1 g : U ′→U we have the naturality square:

DX (U) DX (U ′)

DY (U) DY (U ′)

_◦g

ηU ηU ′

_◦g

.

Hence we have a functor category of diffeological sheaves and natural transformations between them– let us denote it by cShCov{Eucl}, where c comes from concrete and Sh denotes sheaves taken withrespect to the coverage Cov on {Eucl}. We can now state and prove our second important result [1]:Lemma 1.17. Smooth maps between diffeological spaces are in one-to-one correspondence withnatural transformations in the functor category of diffeological sheaves:

cShCov{Eucl}(DX ,DY )∼=C∞(X ,Y ).

Proof. A smooth function between two diffeological spaces f ∈C∞(X ,Y ) gives a natural transforma-tion f̃ : DX ⇒ DY via:

f̃U := f ◦_ : DX (U) 3 φ 7→ f ◦φ ∈ DY (U) ∀U ∈ {Eucl}0,

where f ◦φ ∈DY (U) iff f ∈C∞(X ,Y ) and the naturality square is just the associativity of composition:

f ◦ (φ ◦g) = ( f ◦φ)◦g.

Conversely, a natural transformation η : DX ⇒ DY defines a smooth function η∗ : X → Y when we(again) identify plots from the one-point set with their images, e.g φx : {∗} 7→ x ∈ X with x ∈ X –we will refer to this as the point-map bijection. The function η∗ is then given by taking x ∈ X tothe element in Y corresponding to η{∗}(φx) : {∗}→ Y . We will show that these operations are eachothers inverses.

Notice first that ( f̃ ){∗} takes a plot φx ∈DX{∗} to f ◦φx : {∗} 7→ f (x), and hence ( f̃ )∗(x) = f (x) ∈Y .Since x was arbitrary, the covering condition gives ( f̃ )∗ = f .

For the other direction, notice first that η̃∗ = η holds iff:

η̃∗(ψ)(u) = (η∗ ◦ψ)(u) = η∗(ψ(u)) = ηU (ψ)(u) ∈ Y

for all ψ ∈DX (U) and an arbitrary u∈U , where the first two equalities is just unpacking the definitionof η̃∗(ψ). We then need to show that η∗(ψ(u)) = ηU (ψ)(u). To see why this is true, let us considerthe following naturality square:

DX{∗} DX (U)

DY{∗} DY (U)

_◦g

η{∗} ηU_◦g

,

where g :=!U is the unique function U→{∗}. For the indicating plot of ψ(u)∈X , i.e. φψ(u) ∈DX{∗},the commutativity of the above naturality square gives:

η{∗}(φψ(u))◦g = ηU(φψ(u) ◦g

).

1Notice the change of direction of g comparing to the definition of a natural transformation – this is becausewe are now dealing with contravariant functors.

21

But notice that on the left hand side we have a function that takes u ∈U to the point in Y indicatedby the plot η{∗}(φψ(u)), i.e.:

η{∗}(φψ(u))◦g : U 3 u 7→ η∗(ψ(u)) ∈ Y.

Further, on the right hand side we take ηU of the plot on X that takes u ∈U to φ(u):

φψ(u) ◦g : U 3 u 7→ ψ(u) ∈ X ,

and hence we have ηU(φψ(u) ◦ g

)= ηU (ψ)(u). We then see that η̃∗(ψ)(u) = ηU (ψ)(u) ∈ Y and

since ψ ∈ DX (U) and u ∈U were arbitrary, this finishes the proof.

We have just seen that diffeological sheaves and natural transformations between them correspondto diffeological spaces and their smooth maps. We will now make this statement firmer by statingwhat exactly does it mean for two categories to be essentially the same. Let us first define:

Definition 1.33. We say that two objects C,C′ ∈ C0 are isomorphic, denoted C ∼= C′, iff they areconnected by an isomorphism, i.e. an invertible arrow:

C ∼=C′ ⇔ ∃ { f : C→C′},{g : C′→C} ∈ C1 : g◦ f = IdC & f ◦g = IdC′ .

Remark. The isomorphism of objects in {Set} is a bijection of sets, in {Vect} it is an invertible linearmap, in {Mfld} and {Diffeo} the relevant diffeomorphisms, and so on, i.e. the objects are isomorphiciff they are indistinguishable from the point of view of a given categorical structure.

Definition 1.34. Given two functors, F,H : C → D , we call a natural transformation η : F ⇒ Ha natural isomorphism iff each of its components ηC : F(C)→ H(C) is an isomorphism, i.e. it is in-vertible as an arrow in the functor category and we have F(C)∼= H(C) via ηC for all C ∈ C .Remark. Natural isomorphisms play the role of isomorphisms between functors – since we do notdistinguish between isomorphic objects, we can consider the two functors connected by a naturalisomorphism the same, and hence we will write F ∼= G in such a situation.

We consider two categories equivalent iff there is a relevant of a kind of an isomorphism between them:

Definition 1.35. We say that two categories, C and D , are equivalent, denoted C ∼= D , iff we havea pair of functors:

F : C →D , G : D → C

that are inverses to each other up to a natural isomorphism, which means precisely that we have:

F ◦G∼= IdD , G◦F ∼= IdC .

Let us now state the most important result of this section, which summarizes most of the effort thatwe have made so far, and prove it using the last two Lemmas:

Theorem 1.18. The category of diffeological spaces and smooth maps is equivalent to the functorcategory of diffeological sheaves:

{Diffeo} ∼= cShCov{Eucl}.

Proof. In the light of the Lemma 1.16, we define the functor F : {Diffeo}→ cShCov{Eucl} to be theone that takes a diffeological space (X ,DX ) to a pre-sheaf DX on the Euclidean site that assignsthe plots U → X to the open subsets U and the pre-composition operation to the smooth maps

22

between them. The smooth maps between diffeological spaces are, of course, taken to the naturaltransformations vie the construction from the Lemma 1.17. The first functor then looks like this:

F : {Diffeo}→ cShCov{Eucl}(X ,DX ) 7→ DX ,

{ f : X → Y} 7→ f̃ := f ◦_ : DX ⇒ DY .

Let us define the inverse functor G : cShCov{Eucl}→ {Diffeo} to take the sheaf DX to a diffeologicalspace formed on its image on the one-point set with the disjoint union of all the plots as DX , anda natural transformation to the corresponding functions as in the Lemma 1.17.:

G : cShCov{Eucl}→ {Diffeo}

DX 7→(DX{∗},

⊔U∈{Eucl}0

DX (U)),

{η : DX ⇒ DY} 7→ {η∗ : X → Y}.

The claim is now that G ◦ F ∼= Id{Diffeo} and F ◦G ∼= IdcShCov{Eucl}. Notice here, that from theLemma 1.16 it follows that replacing the diffeological space with the corresponding sheaf and backgives the same diffeological space, i.e. we always have that G◦F(X ,DX )∼= (X ,DX ), and similarlyF ◦G(DX ) ∼= DX . Let us denote these isomorphisms by ξ(X ,DX ) and ρDX , respectively. Further,in the Lemma 1.17 we have shown that ( f̃ )∗ = f and η̃∗ = η , which gives the commutativity of thenaturality squares for ξ : G◦F ⇒ Id{Diffeo} and ρ : F ◦G⇒ IdcShCov{Eucl}:

G◦F(X ,DX ) G◦F(Y,DY )

(X ,DX ) (Y,DY )

ξ(X ,DX )

G◦F( f )

ξ(Y,DY )f

,

F ◦G(DX ) F ◦G(DY )

DX DY

ρDX

F◦G( f )

ρDYη

,

and hence G◦F and F ◦G are indeed naturally isomorphic to the identity functors.

Remark. From now on, we will use those two equivalent definitions interchangeably, i.e. a diffeologi-cal sheaf might be referred to as a diffeological space or a natural transformation as a smooth map,along the line of the equivalence just described.

We have thus shown that the theory of diffeological spaces is actually a theory of concrete sheaveson the Euclidean site. We find this very interesting, for various reasons. Firstly, as we will shortly see,from this perspective the origin of some of the nice categorical properties of {Diffeo} – namely theexistence of the diffeological structure on all kinds of derived spaces – can be understood. Secondly,as seen by many to be the main purpose of speaking about mathematical structures in categoricalterms, it allows for generalizations: we could now change the underlying site for something moregeneral1 or simply different,2 forget about sets and just deal with abstract, not necessary concretesheaves3 etc., and as long as we keep to the sheaf on a site framework, there is a fairly good chancethat the categorical properties will continue to hold.

1A friend of mine, Nesta van der Shaaf, is aiming for the category of super cartesian spaces.2We are wondering, what would be the result of restricting to the one-dimensional plots, like for Frölicher

spaces, by taking the site to be formed on the first two Euclidean spaces: the one-point set {∗} and R (plus theempty set), with the same coverage.

3This is the idea of a smooth set [17].

23

1.2.4 Sheaf-theoretic origin of categorical properties

In this paragraph we are going to show how the fact that we have natural diffeological structureon different kinds of derived spaces can be understood as a consequence or in the light of thesheaf-theoretic perspective.

Limits and colimits

Let us first broaden a bit more our categorical language by introducing the concepts of limits andcolimits, which we will need not only to understand why products, coproducts, fibered products,pushout sets, one-point set and an empty set come naturally equipped with inherited diffeologicalstructures, but also for the definitions of the tangent spaces in the coming section.

The categories that we have seen so far were all very rich in the sense of having many objects andarrows. However, it is useful to realize that simple diagrammatic structures can also be thoughtas categories. For example, a one object category with the identity arrow alone is a category. For twoobjects, we have a discrete category – the one with the identity arrows alone, the indiscrete category,where we have all the possible arrows, i.e. also an arrow between the objects in both directions andtwo other categories with one of the non-identity arrows missing, etc.1 The functors from such simplecategories can then be thought of as mimicking their shape in the target category:

Y :

•

•

• •

7→

A

B

C D

f

h l

g

.

We will thus refer to a functor2 Y : I → C as a diagram of shape I in C . It is a very useful pictureto keep in mind, especially because we will now be actually interested in such simple diagrams only.Let us now define:Definition 1.36. A cone for a diagram Y : I → C is a natural transformation η : ∆C⇒ Y, where∆C : I → C is the constant functor that sends every object of I to C ∈ C and each arrow to IdC.Remark. A cone for Y can be then represented as a diagram in C with the object C on the top andan arrow C→ Y0(i) for any object i ∈I0, such that we have:

C

Y0(i) Y0( j)

for all the arrows Y0(i)→ Y0( j) in the image of Y1. This justifies the terminology.

In general, we have multiple cones for a given functor, i.e. for a diagram of a given shape. However,there might be a very special one:Definition 1.37. A cone for a diagram Y : I → C is called limiting, denoted ηlim : ∆lim I ⇒ Y ifffor any other cone η : ∆C⇒ Y we have a unique arrow C→ limY such that for all i ∈I0 we have:

C limY

Y0(i)

.

1We can of course always add as many arrows as we want.2Usually the category I is required to be small, but those are the only categories we are considering (see the

Definition 1.18).

24

This is referred to as the universal property of a limiting cone.

Definition 1.38. A limit of a diagram Y : I → C , denoted limY, is the vertex of the limiting cone.

Remark. If we had two such limiting cones, we would have an arrow between their vertices bothways, and hence they need to be isomorphic, and because of the above commutativity requirementthe cones coincide. Notice also, that such limiting cones do not need to exist.

As usually with the category-theoretic notions, we also have the dual one – that of a colimit, and justlike with the covariant and contravariant functors, it’s all about the direction of the arrows. Keepingthe notation as above, we define:

• the cocones the same way as we defined cones with the arrows now pointing into C,

• the limiting cocone to be the one that comes with an arrow colimY→C and

• the colimit of a functor to be its vertex.

In other words, colimits are limits in the opposite category, and vice versa.

The concepts of a limit and colimit are useful for us at this point because they allow for a generalizationof notions such as products, direct sums, pull-back sets and fibered products from the category of setsand functions to an arbitrary one. We will point out some examples of the relevant constructions,leaving some of the details to the reader. Let us first consider a functor from the discrete 2-objectcategory to {Set}:

Y× : {• •} 7→ {S S′}

A cone ∆P⇒ Y× for such a diagram is then of the form:

P

S S′

f g ,

i.e. we have a pair of functions from f : P→ S and g : P→ S′. The universal property of the limitingcone means that for such a cone we always have a unique function h : P→ limY× such that:

P

limY×

S S′

f gh

πS π ′S

,

i.e. any pair of functions to S and S′ uniquely factors through the set limF , which is the distinguishingproperty of the product S×S′. Indeed, it is the only set for which such a function, given by:

h := f ×g : P 3 p 7→ ( f (p),g(p)) ∈ S×S′,

always exists and is unique. We then have limY× = S×S′. With a similar argument one can showthat the colimit of this functor is the disjoint union of S and S′, i.e. colimY× = StS′. It is also easyto see that this construction generalizes to finite products and disjoint unions, which for the reasonthat should be clear by now will be referred to as coproducts.

25

To see how the fibered products and pushout sets also arise as limits and colimits, let us consider thefollowing functor:

Yy :•

• •

7→S′

S B

g

f

.

A cone for such a diagram is of the following form:

P S′

S B

i

j

g

f

,

and the universal property for the limiting one means that for any diagram as above we have a uniquearrow h : P→ limYy such that:

P

limYy S′

S B

i

j

h

pS

pS′g

f

This is in turn a distinguishing property of a fibered product, with projections πS and π ′S coming fromthe product projections π1 and π2. We then have limYy = S×B S′. Similarly, the pushout set arisesas a colimit of this functor, i.e. we have colimYy = StB S′.

Notice also that the limit of a functorY• : {•} 7→ S,

is, regardless the chosen set S, the one-point set limY• = {∗}. Indeed, it is the only set for which wealways have a unique map from any other, while the the colimit is always the empty set: colimY• = /0for a similar reason. We then also recover the initial and final objects of {Set} as a colimit and a limit,respectively.

Thus, we generalize products, coproducts, fibered products and pushouts to an arbitrary categoryby identifying them with the, existing or not, limits or colimits of the corresponding diagrams.

We have already seen that all the above sets that can be constructed as limits or colimits of diagramsin {Set} can be equipped with the inherited diffeological structures, and this is not a coincidence –in fact, it can be shown that the category cShCov{Eucl} is complete and cocomplete, meaning thatit admits all finite1 limits and colimits [1] – the existence of the inherited diffeological structureson the products, coproducts, fibered products, pushout sets, one-point set and the empty set canbe simply seen a consequence of this fact. The completeness and cocompleteness properties aregenerally true for sheaf categories, and the existence of the inherited diffeological structures on thosesets can be seen as a consequence of {Diffeo} being a sheaf category at heart. Indeed, the existenceof the relevant object in a sheaf category is basically a consequence of its existence in {Set}. Thereare some details we will skip here, but as an enlightening illustration of this fact, let us considera simpler case of a pre-sheaf category:

1Finite refers to the cardinality of the object set of the domain of functors we consider, i.e. we have limitsand colimits to all of the diagrams that we previously referred to as simple

26

Lemma 1.19. Limits and colimits in pre-sheaf categories can be computed point-wisely, meaningthat for a diagram Y : I → PSh(C ), we have:

(limY)(C) = lim(Y(C)) ∀C∈C , colim(Y)(C) = colim(Y(C)) ∀C∈C

i.e the (co)limiting object for Y is a pre-sheaf on C that takes an object C ∈ C to the set constructedas a (co)limit of a diagram in {Set} that arises by evaluating a diagram Y on C ∈ C .

Proof. Lets consider a cone η : ∆D⇒ Y with a vertex D ∈ PSh(C ). For each arrow {a : i→ j} ∈I1we then have a commutative trangle of natural transformations of the form:

D

Y(i) Y( j)

ηi η j

Y(a)

,

where Y(i),Y( j) ∈ PSh(C ). Evaluating this diagram on an object C ∈ C gives a diagram in {Set}:

D(C)

Y(i)(C) Y( j)(C)

(ηi)C (η j)C

Y(a)C

.

Let us now take the limiting cone ξ (C) : ∆limY(C)⇒ Y(C) in {Set}, composed from the trianglesas above. The universality property then says that we have a unique function h(C) : lim(Y(C))→D(C)such that for each {a : i→ j} ∈I1 we have a diagram in {Set}:

D(C)

lim(Y(C))

Y(i)(C) Y( j)(C)

h(C)(ηi)C (η j)C

ξ (C)i ξ (C) j

Y(a)C

We can then define a presheaf (limY)(C) := limY(C) and a natural transformation h : D⇒ (limY)(C)point-wisely, i.e. via hC := h(C) and similarly ξi : (limY)(C)⇒ Y(i) via (ξi)C := ξ (C)i. It is thenthe unique one for which in the sheaf category we have:

D

limY

Y(i) Y( j)(C)

hηi η j

ξi ξ j

Y(a)

An exactly analogous proof works for colimits.

We then see that the existence of limits and colimits in pre-sheaf categories is a direct conse-quence of their existence in {Set}. However, if we want to apply this construction to our categorycShCov{Eucl} things get a little more tricky – while limits of concrete sheaves are again concrete

27

sheaves,1 colimits might not be. However, there are operations of sheafification and concretization2

that take the resulting pre-sheaves to the elements of cShCov{Eucl} that are limits of the relevantdiagrams in there. Since we want to keep our presentation elementary, instead of going into detailshere we will show how this works on some examples – we will construct the product and the terminalobject – and refer an interested reader to [1].

Lets consider a functor Y as before but with cShCov{Eucl} as its codomain:

Y× : {• •} 7→ {DX DY},

which is now a diagram in {Diffeo}. A cone for this diagram is of the form:

DZ

DX DY

η ξ ,

where DZ is an arbitrary diffeological sheaf with natural transformations to DX and DY . Followingthe prescription above, we define the limiting sheaf as follows:

DX ×DY : {Eucl}op→{Set}U 7→ DX (U)×DY (U),

{ f : U ′→U} 7→ {DX ( f )×DY ( f ) : DX (U)×DY (U)→ DX (U ′)×DY (U ′)}

Let us now check that the universal property holds and that this is the diffeological space we expect:

Lemma 1.20. The functor DX ×DY is a limit of Y in cShCov{Eucl} and corresponds to the productX×Y equipped with the product diffeology, i.e. we have:

limY× = DX ×DY ∼= DX×Y .

Proof. Notice first that since the underlying set is given by evaluation on the terminal object, we have:

DX ×DY{∗}= DX{∗}×DY{∗} ' X×Y.

Moreover, evaluating the limiting cone on {∗} by definition gives a limiting cone in {Set}:

X×Y

X Y

πX πY ,

and hence, because of the bijection of the Lemma 1.17, the natural transformations of the limitingcone in {Diffeo} are given by:

πX ◦_ : DX ×DY ⇒ DX , πY ◦_ : DX ×DY ⇒ DY .

1The sheaf property is a consequence of the fact that the inclusion ShCov(C ) ↪→ PSh(C ) is a left adjoint, theproof of the concreteness property can be found in [1].

2One can see the effect of those operations in the definition of a coproduct diffeology, where we need to dealwith the connected components of the open subsets separately.

28

The universal property means here that we should always have a unique natural transformationρ : D⇒ DX ×DY such that:

DZ

DX×Y

DX DY

ρ

η ξ

π1◦_ π2◦_

.

But again, ρ is given by a composition with a function h : Z→ X ×Y , and similarly η = f ◦_ forf : Z→ X and ξ = g◦_ for g : Z→Y . Hence, evaluating the above diagram on {∗} gives that indeedthere is always a unique such ρ , namely we need to take h := f ×g.

Further, since plots are the same things as smooth parametrizations, taking (Z,DZ) to be an opensubset of a Euclidean space equipped with the manifold diffeology, the universality property, onceagain evaluated on {∗}, gives us a unique plot φ : U → X ×Y for any pair of plots φX ∈ DX (U)and φY ∈ DY (U) such that πX ◦φ = φX and πY ◦φ = φY . Conversely, since composition of smoothmaps is smooth, any plot on X×Y defines such a pair of plots on X and Y and hence all of the plotson X×Y arise in this way.

We then see, that the sheaf thus defined is the same as the one given by the product set equipped withthe product diffeology, i.e. DX×DY 'DX×Y . From the Lemma 1.16 it then follows that it is an objectin cShCov{Eucl}, which finishes the proof.

Let us now see how a terminal object of {Diffeo}, i.e. the one-point set with all the parametrizationssmooth, arises in a similar manner. Notice first, that the corresponding diffeological sheaf is of theform:

D{∗} : {Eucl}op→{Set}U 7→

{!U : U →{∗}

},

{ f : U ′→U} 7→ {!U 7→!U ◦ f =!U ′},

where with "!" we denoted the unique functions taking all of the points in the domain to the onlyelement of {∗}. But since the one-point sets are all isomorphic, it is the same thing as a constantsheaf D{∗} = ∆{∗}. Notice now, that when we consider the functor:

Y• : {•} 7→ DX ,

the point-wisely calculated limit takes every object to the limit in {Set}, i.e. the one-point set andwe have ∆{∗} = limY•.

Similar analysis can be done to show that the diffeological structure on fibered products is alsoa consequence of a sheaf-theoretic structure of {Diffeo} and the fact that those are spaces that ariseas limits in {Set}. As mentioned before, colimits arise in a similar manner – we just need to makesure in the end that the resulting pre-sheaves are concrete sheaves. Hence, the origin of the naturaldiffeological structure on disjoint unions, pushout sets and an empty set can be recognized to be thefact that {Diffeo} is a sheaf category at heart plus that we have the sheafification and concretizationfunctors with some nice structure-preserving properties.1

1See [1].

29

Cartesian closedness

Except for the inherited diffeological structures on the sets that arise as limits and colimits in {Set},we have also seen that the category {Diffeo} is closed, i.e. it admits an exponential object for anypair of diffeological spaces (X ,DX ) and (Y,DY ) given by the diffeological space of smooth mapsC∞(Y,X) equipped with the functional diffeology. We then would like to see how C∞(Y,X) arisesparallelly to the exponential object in {Set}. Let us recall:

In {Set}, the exponential object of S and T is the set of functions T → S denoted by ST = {Set}(T,S).It comes naturally equipped with the evaluation map:

{ f : P→ ST} 7→ ev( f ) : (p, t) 7→ f (p)(t) ∈ S,

which provides a bijection {Set}(P,ST ) ∼= {Set}(P×T,S) that we can treat as a defining propertyof the set ST – clearly, this is the unique set with such a property, and it always exists. We willnow use the coresponing version of this isomorphism to define the sheaf DDXY and then show thatit is indeed the diffeological space of smooth maps with functional diffeology:

Lemma 1.21. We have a unique object DDXY ∈ cShCov{Eucl}0 for which the following holds:

cShCov{Eucl}(DZ ,DDXY )∼= cShCov{Eucl}(DZ×DX ,DY ). (∗)

Moreover, the isomorphism above is necessarily given by the set-theoretic evaluation map appliedto the functions corresponding to the natural transformations, i.e. we have:

cShCov{Eucl}(DZ ,DDXY ) 3 η = (η∗ ◦_) 7→ (ev(η∗)◦_) ∈ cShCov{Eucl}(DZ×DX ,DY ).

Further, the diffeological space corresponding to DDXY is isomorphic to the space of smooth func-tions between X and Y equipped with the functional diffeology, i.e. we have an isomorphismin cShCov{Eucl}:

DDXY ∼= (C∞(X ,Y ),DC∞(X ,Y )).

Before we proceed to the proof, let us recall a few facts about the one-point set and show how theygeneralize to the corresponding facts concerning the terminal object ∆{∗} ∈ cShCov{Eucl}. Noticenow, that the point-map bijection that we have seen and used, i.e. the correspondence between theconstant functions φx : {∗} 7→ x ∈ X and the elements x ∈ X that we have seen, gives us the followingbijection:

{Set}({∗},X)∼= X .

Similarly, an object in a category can be identified with the set of arrows pointing at it from theterminal object – this is generally true for concrete categories1, and especially in cShCov{Eucl}.We then have2 that for any DX ∈ cShCov{Eucl}0:

cShCov{Eucl}(∆{∗},DX )∼= DX .

Indeed, any natural transformation η : ∆{∗}⇒ DX picks an element of DX (U) for each U ∈ {Eucl}0in a way compatible with composition with smooth functions between them, and considering allsuch natural transformations gives the whole DX , and thus we recover the diffeological structureof (X ,DX ). Moreover, in {Set} we also have:

{∗}×X ∼= X ,1In fact morphisms from the terminal object generalize the notion of an element of an object from {Set}

to arbitrary categories (admitting a terminal object).2This is in fact an incarnation of the Yoneda lemma.

30

which holds more generally as well. In our case, the presheaves DX and ∆{∗}×DX agree point-wisely,precisely because of the above isomorphism, and hence they define the same diffeological spaces, i.e.for any DX ∈ cShCov{Eucl}0 we also have:

∆{∗}×DX ∼= DX .

Equipped with this very useful observations, we are now ready to prove the Lemma 1.21:

Proof. Using the above observations, setting DZ = ∆{∗} in (∗) gives:

DDXY ∼= cShCov{Eucl}(∆{∗}×DX ,DY )∼= cShCov{Eucl}(DX ,DY )'C∞(X ,Y ),

which was one of our claims. Notice however, that by now the last equivalence, denoted by "'" is justthe set-theoretic one – we also want to show that the diffeology on DDXY and C

∞(X ,Y ) agree. But letus first take a closer look at the isomorphism (∗). Notice, that since the above holds, we have that theunderlying set of DDXY is contained in the set (isomorphic to the set) of functions Y

X . Now, consideringa general DZ again, in the light of the Lemma 1.17, a natural transformation η : DZ ⇒ DDXY hasto be of the form η = η∗ ◦_ where η∗ : Z→Y X . Similarly, since we have DZ×DX ∼= DZ×X , a naturaltransformation ξ : DZ×DX ⇒DY is of the form ξ = ξ∗ ◦_ for ξ∗ : Z×X →Y . If we now take a pairof constant plots:

φz : {∗} 7→ z ∈ Z, φz,y : {∗} 7→ (z,y) ∈ Z×Y,

the isomorphism (∗) relates η(φz) = φz ◦η∗, which corresponds to η∗(z) ∈ Y X via the point-mapbijection, to ξ (φz,x) = φz,x ◦ξ∗, which in turn corresponds to ξ∗(z,x) ∈ Y , and hence we can identifyξ∗(z,x) = η∗(z)(x). The (∗) is then indeed given thr